The global well-posedness of equations (REF ) and (REF ) for
fixed positive \(b>0\)  in the case of homogeneous Dirichlet boundary conditions
and small, smooth initial data has been studied in [1]}, [2]} and [3]}, [4]}, respectively. In the
inviscid case with \(b=0\) , smooth solutions can be expected to exist only locally in time. This
observation has been made rigorous in [5]} for the Westervelt equation. The Kuznetsov and Westervelt equations with \(b>0\)  have also been analyzed in [6]} and [7]}, respectively, in the limit of third-order acoustic models in time for a vanishing thermal relaxation parameter.
